Abstract

Let $G$ be a compact group. Let $\sigma$ be a continuous involution
of $G$. In this paper, we are
concerned by the following functional equation
$$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^{-1})\,dt=2g(x)h(y), \quad
x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be
determined, are complex continuous functions on $G$ such that $f$ is
central. This equation generalizes d'Alembert's and Wilson's
functional equations. We show that the solutions are expressed by
means of characters of irreducible, continuous and unitary
representations of the group $G$.